Laplace approximation of Lauricella functions F A and F D
The Lauricella functions, which are generalizations of the Gauss hypergeometric function 2 F 1, arise naturally in many areas of mathematics and statistics. So far as we are aware, there is little or nothing in the literature on how to calculate numerical approximations for these functions outside t...
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| Format: | Article |
| Language: | English |
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Springer
2015
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| Online Access: | https://eprints.nottingham.ac.uk/29139/ |
| _version_ | 1848793723495776256 |
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| author | Butler, R.W. Wood, Andrew T.A. |
| author_facet | Butler, R.W. Wood, Andrew T.A. |
| author_sort | Butler, R.W. |
| building | Nottingham Research Data Repository |
| collection | Online Access |
| description | The Lauricella functions, which are generalizations of the Gauss hypergeometric function 2 F 1, arise naturally in many areas of mathematics and statistics. So far as we are aware, there is little or nothing in the literature on how to calculate numerical approximations for these functions outside those cases in which a simple one-dimensional integral representation or a one-dimensional series representation is available. In this paper we present first-order and second-order Laplace approximations to the Lauricella functions F(n)A and F(n)D. Our extensive numerical results show that these approximations achieve surprisingly good accuracy in a wide variety of examples, including cases well outside the asymptotic framework within which the approximations were derived. Moreover, it turns out that the second-order Laplace approximations are usually more accurate than their first-order versions. The numerical results are complemented by theoretical investigations which suggest that the approximations have good relative error properties outside the asymptotic regimes within which they were derived, including in certain cases where the dimension n goes to infinity. |
| first_indexed | 2025-11-14T19:04:50Z |
| format | Article |
| id | nottingham-29139 |
| institution | University of Nottingham Malaysia Campus |
| institution_category | Local University |
| language | English |
| last_indexed | 2025-11-14T19:04:50Z |
| publishDate | 2015 |
| publisher | Springer |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | nottingham-291392018-01-19T05:41:00Z https://eprints.nottingham.ac.uk/29139/ Laplace approximation of Lauricella functions F A and F D Butler, R.W. Wood, Andrew T.A. The Lauricella functions, which are generalizations of the Gauss hypergeometric function 2 F 1, arise naturally in many areas of mathematics and statistics. So far as we are aware, there is little or nothing in the literature on how to calculate numerical approximations for these functions outside those cases in which a simple one-dimensional integral representation or a one-dimensional series representation is available. In this paper we present first-order and second-order Laplace approximations to the Lauricella functions F(n)A and F(n)D. Our extensive numerical results show that these approximations achieve surprisingly good accuracy in a wide variety of examples, including cases well outside the asymptotic framework within which the approximations were derived. Moreover, it turns out that the second-order Laplace approximations are usually more accurate than their first-order versions. The numerical results are complemented by theoretical investigations which suggest that the approximations have good relative error properties outside the asymptotic regimes within which they were derived, including in certain cases where the dimension n goes to infinity. Springer 2015-12 Article PeerReviewed application/pdf en https://eprints.nottingham.ac.uk/29139/1/L-M_4_Rev_1%20%283%29.pdf Butler, R.W. and Wood, Andrew T.A. (2015) Laplace approximation of Lauricella functions F A and F D. Advances in Computational Mathematics, 41 (6). pp. 1015-1037. ISSN 1572-9044 Gauss hypergeometric function; Lauricella functions; vector- argument hypergeometric functions http://link.springer.com/article/10.1007/s10444-014-9397-5 doi:10.1007/s10444-014-9397-5 doi:10.1007/s10444-014-9397-5 |
| spellingShingle | Gauss hypergeometric function; Lauricella functions; vector- argument hypergeometric functions Butler, R.W. Wood, Andrew T.A. Laplace approximation of Lauricella functions F A and F D |
| title | Laplace approximation of Lauricella functions F A and F D |
| title_full | Laplace approximation of Lauricella functions F A and F D |
| title_fullStr | Laplace approximation of Lauricella functions F A and F D |
| title_full_unstemmed | Laplace approximation of Lauricella functions F A and F D |
| title_short | Laplace approximation of Lauricella functions F A and F D |
| title_sort | laplace approximation of lauricella functions f a and f d |
| topic | Gauss hypergeometric function; Lauricella functions; vector- argument hypergeometric functions |
| url | https://eprints.nottingham.ac.uk/29139/ https://eprints.nottingham.ac.uk/29139/ https://eprints.nottingham.ac.uk/29139/ |